428 XV NORMED ALGEBRAS AND SPECTRAL THEORY B = (7+ 7T*)-1 and if for each y e E we put x = B • y, show that the hypotheses imply that y e dom(T*), by using the fact that x e dom(r*).) 12. Let TV be a normal operator on E. If dom(Nz) — dom(N), show that N is continuous. (If not, there would exist a closed subspace F of E, such that F is the Hilbert sum of an infinite sequence (Fn) of nonzero subspaces stable under N and N*, such that the restriction Nn of N to Fn is a continuous operator and such that for xn e Fn , where the sequence (an) is increasing and tends to +00.) 13. EXTENSIONS OF HERMITIAN OPERATORS (15.13,1) Let E be a separable Hilbert space. A not necessarily bounded operator H on E is said to be hermitian if: (1) dom(H) is dense in E; (2) dom(H) c domCff*), and the restriction of H* to dom(H) is equal to H; in other words, for all x, y in dom(H)9 (15.13.1.1) (H-x\y) = (x\H-y). In particular, (H • x \ x) is real for all x e dom(H). It should be noted that in general H* is not hermitian. When H is con- tinuous, this definition agrees with that of (11.5), and a continuous hermitian operator is the same thing as a continuous self-adjoint operator. An unbounded self-adjoint operator is always hermitian, but as we shall see below there exist closed hermitian operators which are not self-adjoint. Example (15.13.2) Let Q) be the vector space of indefinitely differentiable complex- valued functions on R with compact support. If /? denotes Lebesgue measure on R, then @ may be identified with a subspace of L£(R, ft) (13.19), which is dense in L£(R, /?) (17.1.2). It follows immediately from the formula for integration by parts (8.7.5) and the fact that the functions belonging to ^ vanish outside a compact set, that for all x9 y in ^ = f Dx(t)XT) dt=- f x(ODXF) dt = -(x \ Dy). (D* | y) This may be expressed by saying that iD = H is a hermitian operator. The conclusion is the same if we replace L£(R, ft) by Lc(I, /?), where I is a (boundedrem and the